
As indicated by the name, the SGF is a generating function of the \(\lbrace 6j\rbrace \) -symbols, thus it contains all the information of the spin network evaluation of a tetrahedron graph in its expansion.
That the spins are restricted to be half-integers and admissible in the \(\lbrace 6j\rbrace \) -symbols brings difficulties in numerical analysis, which was often circumvented by looking at the large \(j\)  limit of the \(\lbrace 6j\rbrace \) -symbol.
The computation issues are translated to complex analysis when we consider the SGF, which is expected to be easier to deal with as it is a continuous function of 12 independent spinors. The large \(j\)  limit of the \(\lbrace 6j\rbrace \) -symbols then contributes dominantly at the poles of the SGF [1]}.
Other than the possible computational benefits, the SGF is attractive because of its remarkable closed form. A similar closed form for the generating function of the \(9j\) -symbols was also constructed in the original work by Schwinger [2]}. Later on, Bargmann [3]} reproduced Schwinger's results and his method was used to construct the generating function of the \(12j\) - and \(15j\) -symbols [4]} and more generally the \(\lbrace 3nj\rbrace \) -symbols [5]}, which can also be expressed as loops.

The SGF is also understood as (the evaluation of) a quantum state - the scaleless spin network state -
that describes the physical state of a tetrahedron graph in the spinor representation.
The physical state of a graph is most simply represented by the flatness projection written in terms of the holonomies, which is the product of the Dirac deltas on holonomies corresponding to independent cycles in the graph. For a graph embedded on the manifold of trivial topology, the physical state is unique, which is the case for the tetrahedron graph embedded in the 2-sphere.
Using the notation in fig.REF , the physical state for a tetrahedron graph can be written as
\(\psi _{\text{phys}}(g_l)=\delta (g_6g_1g_2^{-1})\delta (g_1^{-1}g_5^{-1}g_3^{-1})\delta (g_3g_4^{-1}g_2)\,.\) 
When projected onto the scaleless spin network basis, \(\psi _{\text{phys}}(g_l)\)  gives exactly the SGF,
\(\psi _{\text{phys}}(\lbrace z_l,\tilde{z}_l\rbrace )= \int _{\mathrm {SU}(2)^6}\prod _{l=1}^6 \mathrm {d}g_l {\mathcal {S}}_\text{tet}^{\lbrace z_l,\tilde{z}_l\rbrace }(g_l) \psi _\text{phys}(g_l) = {\mathcal {S}}(\lbrace z_l,\tilde{z}_l\rbrace )\,.\) 
This can be deduced from (REF ) and the fact that the projection of the physical state onto the spin network basis (REF ) is the \(\lbrace 6j\rbrace \) -symbol [6]}:
\(\psi _{\text{phys}}(j_l)=\int _{\mathrm {SU}(2)^6}\prod _{l=1}^6 \mathrm {d}g_l \, s_\text{tet}^{\lbrace j_l,\iota _n \rbrace }(g_l) \psi _\text{phys}(g_l) = \left\lbrace  \begin{array}{ccc}j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6\end{array} \right\rbrace  \,.\) 
The scaleless spin network states can be used to describe the kinematical information in the Hilbert space of quantum geometry. By the continuous nature of the spinor arguments in a scaleless spin network state, the dynamics given by the Wheeler de-Witt equation would be translated into a differential equation of the SGF [7]}, rather than a recursion relation as for the \(\lbrace 6j\rbrace \) -symbols when we consider the spin network states [6]}.
The interested reader will find a short review with details of this approach in Appendix .

